Arrow Trajectories: Arrow Flight in the Real World

One of my biggest motivations to understand the physics of arrow flight was to model the trajectory of an arrow under real shooting conditions. This means calculating the motion of an arrow under the influence of aerodynamic drag, at any shooting height, and at any launch angle. In this post I will develop a full physical model for arrow flight using the same finite difference approach used in computing arrow acceleration from the bow. As will be shown below, it is easy to solve the differential equations for arrow trajectory in a no drag world. In the real world this is more difficult because the drag coefficient is a function of velocity and velocity is a function of drag. Computers make these calculations easier because we can divide arrow flight into small, finite, intervals of time or distance and then repeat the calculations many times over to model full arrow flight. The calculations are easily implemented in Excel as shown in the example spreadsheet attached to the end of this post.

A good place to start is with the standard trajectory calculation for an arrow in a frictionless world. This calculation is based on Newton's laws of motion which relate acceleration, velocity, and time to

give an exact solution for arrow position Xt, Yt for any time (t). Theta is the launch angle (positive or negative) relative to a line from the nock point to the target, and g is the gravitational acceleration of the earth (9.8 m/ss). The example spreadsheet includes the calculated trajectory based on the above equations. Notice that these calculations do not include any term for aerodynamic drag on the arrow, and we will now address this limitation. In all the calculations in this post I will use metric, SI, units. The attached spreadsheets provide the unit conversions and for those that still like traditional units, remember that a yard is a little less than a meter. Velocity in meters per second are about one third of velocities in feet per second.

An arrow moving through the air experiences a deceleration, drag, due to friction of air on the arrow tip, shaft, and fletching. A typical arrow flies at over 150 mph so think about putting your hand out the window of a performance race car and you get a sense of the drag force on the arrow. The force you experience is related to the shape of the object, viscosity of the air (think about putting your hand in the water from a speeding boat), and the cross sectional area of the object. See Lavander's post for more details. Force is related to mass and acceleration

F = ma

and for drag we usually write this as

F = md

to distinguish drag, d, from arrow acceleration of the bow, a, or gravitational acceleration, g. Unlike gravitational acceleration, d is not constant. Objects moving at the same speed as the air have zero drag. High speed objects have significant drag. We know from ballistics that for subsonic objects d will be a function of a drag coefficient k times arrow velocity raised to some number close to two. In my drag tests I have found the power of two fits experimental data well, so d is defined as

d = -kV2

where the negative sign indicates that drag is slowing the arrow. Different arrows will have different drag coefficients, but a good starting point is a value of 0.0026 (1/meters). Future posts will describe how to measure the drag coefficient for a particular set of arrows. The drag on the arrow is independent of arrow direction, but gravity, g - the other big force accelerating/decelerating the arrow, only operates in the vertical. Therefore, for convenience we divide the drag into X and Y vector components based on the launch angle theta. Similarly, we divide the velocity of the arrow into the vector components Vx and Vy. In this case theta, the launch angle, is measured relative to the plane of the earth.

Unlike the exact solution provided at the start of this post, it is very difficult to solve the above equations for arrow position because drag and velocity are interrelated. A solution is to calculate the change in arrow position over a small enough time interval that drag, dx and dy, are essentially constant (this assumes that velocity is constant over a small time interval). At a very small time interval of 0.004 seconds this assumption is valid and it is possible to calculate the change in X and Y "exactly". The change in X position is a function of X velocity and drag deceleration. The change in Y position is a function of Y velocity, drag deceleration, and gravitational acceleration toward the earth. Notice that the two acceleration terms in the Y direction are combined because d is a constant over small time intervals.

Since we now know delta X and Y for a time interval of 0.004 seconds it is possible to calculate new values for Vx and Vy in terms of change in position over a change in time. The new total velocity is the vector sum of Vx and Vy.

Finally, using the change in X and Y position is is possible calculate an updated angle for arrow flight. This step is important since the flight angle of the arrow changes a lot over a long shot.

This completes one cycle of the trajectory calculation. The changes in velocity and angle are very small for a small time step. However, velocity and angle change a lot as the cycle is repeated many times:

A) compute drag from initial velocity

B) compute the X and Y vector components of drag and velocity

C) compute the change in X and Y position

D) compute new X and Y velocities

E) compute new arrow flight angle

F) go back to step A for a new trajectory cycle.

The total arrow flight time is the sum of each of the small time steps.

In my simulations I typically perform over 1000 small steps for a total of 4 seconds of flight time. By recording values of X, Y, and time for each time step it is possible to plot a correct arrow trajectory that includes the effect of aerodynamic drag.

As expected, including drag in the trajectory calculations causes the arrow to fly slower and shorter for a given launch angle. Drag effects are big for arrow flight over 20 meters. Since we know the arrow velocity at each time step it is also possible to compute the kinetic energy of the arrow as a function of distance. The arrow loses about 10% of the initial kinetic energy for each 10 meters of arrow flight. This is due to aerodynamic drag on the arrow. This correct model for flight allows the technically oriented archer to ask "what if" questions with quantifiable results.

- How will my arrow flight really change with a faster bow?

- If I misjudge distance by 5 meters (yards) how high or low will my arrow land?

- How will pin placement influence launch angle and arrow flight?

- What are the effects of shooting more aerodynamic arrows, lighter arrows, or faster bows?

- Will my arrow have the required kinetic energy to kill a deer at 40 meters?